重新审视古尔定理
Gull's theorem revisited
摘要：
史蒂夫·古尔（Steve Gull）在他的剑桥大学主页上的未发表的工作中概述了使用傅立叶理论证明贝尔定理的方法。Gull的哲学是，贝尔定理（或者可能是其证明中的一个关键引理）可以被视为经典而非量子计算机分布式计算项目的不可行定理。我们提出他的论点，纠正印刷错误，填补空白。在他的论点中，网络中有两台完全独立的计算机。我们需要三台计算机来填补他的证明中的所有空白：第三台计算机向代表贝尔工作中的两个测量站的两台计算机提供随机数流。我们也可以想象，这台计算机被一台克隆的虚拟计算机所取代，在爱丽丝和鲍勃的每台计算机中生成相同的伪随机数。不管怎样，我们都需要假设两人的身份相同。随机性以i.i.d.的同步实现序列的形式存在。隐藏的变量潜在的，否则确定性的物理序列的试验。Gull的证明只需要第三步：将期望重写为给定隐变量的条件期望的期望。
英文摘要：
Steve Gull, in unpublished work available on his Cambridge University homepage, has outlined a proof of Bell's theorem using Fourier theory. Gull's philosophy is that Bell's theorem (or perhaps a key lemma in its proof) can be seen as a no-go theorem for a project in distributed computing with classical, not quantum, computers. We present his argument, correcting misprints and filling gaps. In his argument, there were two completely separated computers in the network. We need three in order to fill all the gaps in his proof: a third computer supplies a stream of random numbers to the two computers representing the two measurement stations in Bell's work. One could also imagine that computer replaced by a cloned, virtual computer, generating the same pseudo-random numbers within each of Alice and Bob's computers. Either way, we need an assumption of the presence of shared i.i.d. randomness in the form of a synchronised sequence of realisations of i.i.d. hidden variables underlying the otherwise deterministic physics of the sequence of trials. Gull's proof then just needs a third step: rewriting an expectation as the expectation of a conditional expectation given the hidden variables.
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